OFFSET
0,3
COMMENTS
Even-numbered terms a(2k) are A002872: 2,7,31,164,999 ("Sorting numbers"); odd-numbered terms are its binomial transform, A080337. The symmetrical set partitions of {-n,...,-1,0,1,...,n} can be classified by the partition containing 0. Thus we get the sum over k of {n choose k} times the number of symmetrical set partitions of 2n-2k elements. - Don Knuth, Nov 23 2003
Number of partitions of n numbers that are symmetrical and cannot be nested (i.e., include a pattern of the form abab). - Douglas Boffey, May 21 2015
Number of achiral color patterns in a row or loop of length n. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 23 2018
Also the number of self-complementary set partitions of {1, ..., n}. The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. - Gus Wiseman, Feb 13 2019
REFERENCES
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 18.
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.
S. V. Pemmaraju and S. S. Skiena, The New Combinatorica, 2001.
Frank Ruskey, Set Partitions
FORMULA
Knuth gives recurrences and generating functions.
a(n) = Sum_{k=0..t(n)} (-1)^k*A125810(n,k) where A125810 is a triangle of coefficients for a q-analog of the Bell numbers and t(n)=A125811(n)-1. - Paul D. Hanna, Jan 19 2009
From Robert A. Russell, Apr 23 2018: (Start)
a(n) = Sum_{k=0..n} Ach(n,k) where
Ach(n,k) = [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)) + [n<2]*[n==k]*[n>=0].
EXAMPLE
Of the set partitions of 4, the following 7 are invariant under 1->4, 2->3, 3->2, 4->1: {{1,2,3,4}}, {{1,2},{3,4}}, {{1,4},{2,3}}, {{1,3},{2,4}}, {{1},{2,3},{4}}, {{1,4},{2},{3}}, {{1},{2},{3},{4}}, so a(4)=7.
For a(4)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD (same as previous example). The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - Robert A. Russell, Apr 23 2018
From Gus Wiseman, Feb 13 2019: (Start)
The a(1) = 1 through a(5) = 12 self-complementary set partitions:
{{1}} {{12}} {{123}} {{1234}} {{12345}}
{{1}{2}} {{13}{2}} {{12}{34}} {{1245}{3}}
{{1}{2}{3}} {{13}{24}} {{135}{24}}
{{14}{23}} {{15}{234}}
{{1}{23}{4}} {{1}{234}{5}}
{{14}{2}{3}} {{12}{3}{45}}
{{1}{2}{3}{4}} {{135}{2}{4}}
{{14}{25}{3}}
{{15}{24}{3}}
{{1}{24}{3}{5}}
{{15}{2}{3}{4}}
{{1}{2}{3}{4}{5}}
(End)
MATHEMATICA
<<DiscreteMath`NewCombinatorica`; Table[t = SetPartitions[n]; t= t /. Thread[Range[n] -> Range[n, 1, -1]]; t= 1 + RankSetPartition /@ t; t= ToCycles[t]; t= Cases[t, {_Integer}]; Length[t], {n, 7}]
(* second program: *)
QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; Table[cc = CoefficientList[QB[n, q], q]; cc.Table[(-1)^(k+1), {k, 1, Length[cc]}], {n, 0, 30}] (* Jean-François Alcover, Feb 29 2016, after Paul D. Hanna *)
(* Ach[n, k] is the number of achiral color patterns for a row or loop of n
colors containing exactly k different colors *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0],
k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
Table[Sum[Ach[n, k], {k, 0, n}], {n, 0, 30}] (* Robert A. Russell, Apr 23 2018 *)
x[n_] := x[n] = If[n < 2, n+1, 2x[n-1] + (n-1)x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[Ceiling[n/2], k] x[k-Mod[n, 2]], {k, 0, Ceiling[n/2]}],
{n, 0, 30}] (* Robert A. Russell, Apr 27 2018, after Knuth reference *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Mar 15 2003
EXTENSIONS
Offset set to 0 by Alois P. Heinz, May 23 2015
STATUS
approved